Basic Mathematics for College Students

The monomial 7x^3 is called a monomial of third degreeor a monomial of degree 3,

because the variable occurs three times as a factor.

• 5 x^2 is a monomial of degree 2. Because the variable occurs two times
  as a factor: x^2 xx.


•  8 a^4 is a monomial of degree 4. Because the variable occurs four times
  as a factor: a^4 aaaa.


• is a monomial of degree 5. Because the variable occurs five times
  as a factor: m^5 mmmmm.
  We define the degree of a polynomial by considering the degrees of each of its
  terms.

Degree of a Polynomial

The degree of a polynomialis the same as the degree of its term with largest

degree.

For example,

• x^2  5 xis a binomial of degree 2, because the degree of its term with largest
  degree (x^2 ) is 2.


• 4 y^3  2 y7 is a trinomial of degree 3, because the degree of its term with
  largest degree (4y^3 ) is 3.


• is a trinomial of degree 4, because the degree of its term with
  largest degree (3z^4 ) is 4.

1

2 z^3 z

(^4)  2 z 2

1

2

m^5

A-6 Appendix II Polynomials

EXAMPLE (^2) Find the degree of each polynomial:
a. 2 x 4 b. 5 t^3 t^4  7 c. 3  9 z 6 z^2 z^3
StrategyWe will determine the degree of each term of the polynomial.
WHYThe term with the highest degree gives the degree of the polynomial.
Solution
a.Since  2 xcan be written as  2 x^1 , the degree of the term with largest degree is

1. Thus, the degree of the polynomial  2 x4 is 1.
   b.In 5t^3 t^4 7, the degree of the term with largest degree (t^4 ) is 4. Thus, the
   degree of the polynomial is 4.
   c. In 3  9 z 6 z^2 z^3 , the degree of the term with largest degree (z^3 ) is 3. Thus,
   the degree of the polynomial is 3.

Self Check 2

Find the degree of each

polynomial:

a. 3 p^3

b. 17 r^4  2 r^8 r

c.  2 g^5  7 g^6  12 g^7

Now TryProblems 13, 15, and 17

Evaluate polynomials.

When a number is substituted for the variable in a polynomial, the polynomial takes

on a numerical value. Finding this value is called evaluating the polynomial.

2

EXAMPLE (^3) Evaluate each polynomial for x3:
a. 3 x 2 b. 2 x^2 x 3
StrategyWe will substitute the given value for each xin the polynomial and
follow the order of operations rule.